If the `5^(th) and 11^(th)` terms of an A.P. are 16 and 34 respectively. Find the A.P.

Answer

To find the arithmetic progression (A.P.) given that the 5th and 11th terms are 16 and 34 respectively, we can follow these steps:

### Step 1: Define the terms of the A.P.
Let the first term of the A.P. be \( a \) and the common difference be \( d \).

The \( n^{th} \) term of an A.P. can be expressed as:
\[ T_n = a + (n-1)d \]

### Step 2: Write equations for the given terms
From the problem, we know:
- The 5th term \( T_5 = 16 \):
\[ T_5 = a + 4d = 16 \quad \text{(Equation 1)} \]
- The 11th term \( T_{11} = 34 \):
\[ T_{11} = a + 10d = 34 \quad \text{(Equation 2)} \]

### Step 3: Subtract Equation 1 from Equation 2
To eliminate \( a \), we can subtract Equation 1 from Equation 2:
\[ (a + 10d) - (a + 4d) = 34 - 16 \]
This simplifies to:
\[ 10d - 4d = 18 \]
\[ 6d = 18 \]

### Step 4: Solve for the common difference \( d \)
Now, divide both sides by 6:
\[ d = \frac{18}{6} = 3 \]

### Step 5: Substitute \( d \) back into Equation 1 to find \( a \)
Now that we have \( d \), we can substitute it back into Equation 1 to find \( a \):
\[ a + 4d = 16 \]
Substituting \( d = 3 \):
\[ a + 4(3) = 16 \]
\[ a + 12 = 16 \]
Now, subtract 12 from both sides:
\[ a = 16 - 12 = 4 \]

### Step 6: Write the A.P.
Now we have both \( a \) and \( d \):
- First term \( a = 4 \)
- Common difference \( d = 3 \)

The A.P. can be written as:
\[ 4, \, 4 + 3, \, 4 + 2(3), \, 4 + 3(3), \ldots \]
This gives us:
\[ 4, \, 7, \, 10, \, 13, \ldots \]

### Final Answer:
The arithmetic progression is:
\[ 4, 7, 10, 13, \ldots \]